The Fundamentals of MTF, Wiener Spectra, and DQE

Robert M Nishikawa

Kurt Rossmann Laboratories for Radiologic Image ResearchDepartment of Radiology, The University of Chicago

Motivation

Goal of radiology: to diagnosis and treat disease by

Role of Medical Physicist: to help maximize patient benefit while minimizing the cost of the diagnostic imaging study

e.g. diagnostic information vs.. radiation dose

comparison of methods or systems

computed radiography vs. plain film

MRI vs. US

Motivation

Two steps in the radiologic process:

1. image production and display

physical measures (MTF, NPS, NEQ, DQE)

2. image interpretation

observer studies (ROC)

Physical Measures of Image Quality

What is a good (or valid) measure of image quality?

image of a mammogram

series of images(rose 1)

Perceived Image Quality is Proportional to SNR

AQSNR = C

where: SNR = signal-to-noise ratioC = image contrast of the objectA = area of the objectQ = number of quanta per unit area

Outline of Talk

Image Quality Metrics

what are they?

what do they mean?

how are they determined?

Rose Model

Assumptions: (ideal detector)

no blurring

no added noise

perfect absorption of incident quanta

AQSNR = C

Why Work in the Spatial Frequency Domain

performance of a detector depends on the object being imaged

a single analysis in the spatial frequency domain can be used to predict performance of all possible objects

all real objects can be decomposed into sine waves of different amplitudes, frequencies, and phases

computation in spatial frequency domain is easier than in the spatial domain

(multiplication vs. convolution)

Spatial Resolution

can be characterized by limiting resolution

measured using bar pattern

a more complete description is given by modulation transfer function (MTF)

image

rossmann beads and needles

need MTF for intermediate freq; limiting resolution is for high freq only

Outline of Talk

Image Quality Metrics

what are they?

what do they mean?

how are they determined?

Measuring MTF (conceptually)

ImagingSystem

-1

0

1

Input Output

-1

0

1

1.0

0.5

0.01086420

Spatial Frequency (cycles/mm)

measures change in the amplitude of sine waves

MTF Curves

1.0

0.8

0.6

0.4

0.2

0.01086420

Spatial Frequency (cycles/mm)

Measuring MTF (theoretically)a POINT is composed of all spatial frequencies

a LINE is composed of all spatial frequencies in one direction and zero frequency in the other

1-D FFT 1-D MTF

digitize with anarrow slitaperture in one direction

digitizewith a small

circular aperturein two directions

2-D Hankeltransform 2-D MTF

PSF

LSF

Measuring MTF (experimentally)Screen-Film Systems

highexposure

lowexposure

digitizeusing narrow

aperture LSF

LSFH&D

correctionbootstrap

2 LSF curves

FFTMTFexponential

extrapolationof tails of LSF

Measuring MTF (experimentally)Digital Detectors (Pre-Sampled)

testobject

exposeusingdigital

detector

ESF differentiate

FFTMTFexponential

extrapolationof tails of LSF

LSF

Oversampling the LSF

10

8

6

4

2

00.00

False-Positive Fraction

Am

plitu

de

Distance (mm)

Aliasing

1.0

0.8

0.6

0.4

0.2

0.0-6.38

False-Positive Fraction

Mod

ulat

ion

Tra

nsfe

r F

acto

r

Spatial Frequency (cycles/mm)

Aliasing

-1.0

-0.5

0.0

0.5

1.0

-10 -5 0 5 10

Distance

MTF of Digital Detectors

non-isotropic --> 2-D display is necessary

MTF in orthogonal directions can be different

Noise

noise can be characterized by standard deviation in the output image

a more complete description is given by the noise power spectrum

noise image

same standard deviation, but different texture

0.1

2

3

4567

1

2 )

0.12 4 6 8

12 4 6 8

10

Spatial Frequency (cycles/mm)

Measuring NPS (conceptually)

ImagingSystem

Input Output

-101

-101

Measure change in thevariation in the amplitude of sine waves

Measuring NPS (theoretically)

a uniform x-ray exposure contains noise at all spatial frequencies

digitizewith a small

circular aperturein two directions

2-D FFT 2-D NPS

digitizewith a long

narrow aperture in one direction

1-D FFT 1-D NPS

uniformlyexposed images

Measuring NPS (experimentally)

uniformlyexposedimage

digitizewith a long

narrow aperturein one direction

1-D FFT

1-D NPS

correction forlength & width of

scanning aperture,detrend data

H&D Curve1-D NPS in terms offluctuations in x-ray exposure

Measuring NPS (experimentally)Digital Detector

2-D FFT

1-D NPS

1-D NPS in terms offluctuations in x-ray exposure

uniformlyexposedimage

low-frequency detrending

data reduction

characteristic curve

Typical NPS

10-6

2

4

68

10-5

2

4

68

10-4

2m

m 2)

0.12 3 4 5 6 7

12 3 4 5 6 7

102

Spatial Frequency (cycles/mm)

Alternate Methods forMeasuring Noise Power Spectra

Fourier Transform of autocovariance function

analog method

Paradox

Linear Conversion Logarithmic Conversion(Digital Detector) (Screen-Film System)

2.0

1.5

1.0

0.5

0.0100806040200

Pixel Number

1000800600400200

0100806040200

Pixel Number

noise increases with exposure

• noise decreases with exposure

Digital DetectorI = kQdI = kdQ

noise α Q

Solution

Screen-Film Systems

D = G log(Q) + Do

dD = G dlog(Q)

= G log10e dlnQ

= G log10e dQ/Q

noise α (Q)- 0.5

assuming Poisson noise, dQ = √Q

Photon Countingsignal = ∆Q = k∆Q

SNR = ∆Q (Q)- 0.5

= C (Q)0.5

Signal-to-Noise Ratio

Screen-Film Systems

signal = ∆D

= G ∆[log(Q)]

= G log10e ∆Q/Q SNR = ∆Q/Q (Q) 0.5 = C (Q)0.5

where C = ∆Q/Q, the radiation contrast of the object

Signal-to-Noise Ratio

can be characterized

a more complete description is given by NEQ (noise equivalent quanta)

image

CD phantom of digital system

digital low MTF low noise

film high MTF High noise

digital better

0.1

2

3

4567

1

NE

Q (

mm

-2 )

0.12 4 6 8

12 4 6 8

10

Spatial Frequency (cycles/mm)

Measuring NEQ (conceptually)

ImagingSystem

Input Output

-101

-101

Measure change in the mean amplitude and in the variation in the amplitude of sine waves

Noise Equivalent Quanta

0.1 1 10Spatial Frequency (cycles/mm)

10 5

10 4

10 3

NEQ (mm ) -2

Noise Equivalent Quanta (NEQ)

Definition:

=

Q = # of quanta incident on the detector per unit area(assumes unit contrast)

Q DQE( )ωNEQ( )ω

OQ

∆Q2

dQdO

Detective Quantum Efficiency (DQE)

Definition:

DQE(ω) ∆Q 2(ω)

∆O 2(ω)

where ω

∆O2

dd

= spatial frequency

= mean-squared variation in the output

= mean-squared variation in the input

= gain of system

2

Interpretation of DQE

characterizes the efficiency of information transfer from the input to the output of the system

allows comparison to an ideal system

ranges from 0 to 1.0

DQE(ω) = SNRout2 ( )

SNR in2 ( )

SNRout( )

SNR in( )

= SNR in the output image

= SNR incident on the detector

ωω

ω

ω

Interpretation of NEQ

is the number of quanta that an ideal detector would have needed to yield the same SNR

absolute measure of image quality

ranges from 0 to infinity

assumes unit contrast

NEQ( ) = Q DQE( )

For a noise- limited system, SNRin= Q2

NEQ(

ω

) = SNR in( )2ω ω

ω

QdO

How to Calculate DQE (general)

DQE( ) = MTFQ

W( ) ( )2

dOdQ

where = MTF of detector = noise power spectrum of image

= gain of the system

W( )MTF( )

d

ω( )ω

ω

ω

ω

2

d

dQ

How to Calculate DQE(screen-film system)

DQE( ) = γ 2 (log10e)2 MTF

Q W( )

γ dDd(log10Q)

= Qlog10e

dD

dD

Q =

Q

lγ og10e

uu

( )u

u = one dimensional spatial frequency

2

Exposure Dependence

screen-film systems are non-linear

NEQ and DQE are functions of both spatial frequency and x-ray exposure

NEQ = (log10e)2MTF

W(ω Q) ,

(ω Q) ,

(ω)γ 2(Q) 2

H&D curve

DQE 3d

0.05

0.1

0.15

2

0.2

0.25

4 6.56 68 5.5

DQE

SpatialFrequency

(cycles / mm) Log Q

0

Things to Remember

DQE comparisons assume equal SNRin

may not be true: x-ray exposure, kVp

SNRin = C Q

DQE analysis assumes shift-invariant system

DQE & NEQ are measures of SNR

if image is not noise limited, but contrast limited, a system with higher NEQ may not produce a better image

information

ω

Relationship BetweenSNR and NEQ

[SNR = S( ) 2 NEQ( ) d ω ]1/2

ω ω

where is the spatial frequency spectrum of the objectS ( )

Summary

NEQ and DQE are useful parameters for characterizing and understanding medical imaging systems

NEQ and DQE can serve as a basis for comparing different imaging conditions and modalities

NEQ may be useful in furthering our understanding of image perception

Recommended Reading (1) ICRU Report 41: Modulation transfer function of screen-film systems. (2) BRH Report: MTF's and Wiener spectra of radiographic screen-film systems. (3) J. C. Dainty, R. Shaw: Image Science (Academic Press, London, 1974), Chap. 6, 7, and 8. (4) J. S. Bendat, A. G. Piersol: Random Data: Analysis and Measurement Procedures 2nd

edition, (Wiley, New York, 1986). (5) A Rose, Vision: Human and Electronic (Plenum, New York, 1973). (6) C. E. Metz and K. Doi: Transfer function analysis of radiographic imaging systems.

Phys Med Biol 24: 1079 (1979) (7) R. A. Sones, G. T. Barnes: A method to measure the MTF of digital x-ray systems. Med

Phys 11: 166 (1984). (8) H. Fujita, K. Doi, M. L. Giger: Investigation of basic imaging properties in digital

radiography. 6. MTFs of II-TV digital imaging systems. Med Phys 12: 713 (1985). (9) I. A. Cunningham, A. Fenster: A method for modulation transfer function determination

from edge profiles with correction for finite-element differentiation. Med Phys 14: 533 (1987).

(10) M. Dragnova, J. A. Rowlands: Measurement of the spatial Wiener spectrum of nonstorage imaging devices. Med Phys 15: 151 (1988).

(11) J. A. Rowlands, G. DeCrescenzo: Wiener noise power spectra of radiological television systems using a digital oscilloscope. Med Phys 17: 58 (1990).

(12) I. A. Cunningham and B. K. Reid: Signal and noise in modulation transfer function determinations using the slit, wire, and edge techniques, Med Phys 19(4):1037-1044, 1992.

(13) J.M. Sandrik, R.F. Wagner, Absolute measures of physical image quality: Measurement & application to radiographic magnification, Med. Phys. 9: 540(1982).

(14) R.M. Nishikawa, M.J. Yaffe, Signal-to-noise properties of mammographic film-screen systems, Med. Phys. 12, 32-39 (1985).

(15) PC Bunch, KE Huff, R Van Metter, Analysis of the detective quantum efficiency of a radiographic film-screen combination, J. Opt Soc Am A 4, 902-909 (1987).

(16) J. T. Dobbins, Effects of undersampling on the proper interpretation of modulation transfer function, noise power spectra, and noise equivalent quanta of digital imaging systems, Med Phys 22, 171-81 (1995).

(17) J. T. Dobbins, D.L. Ergun, L. Rutz, et al., DQE(f) of four generations of computed radiography acquisition devices, Med Phys 22, 1581-1593 (1995).

(18) M. L. Giger and K. Doi, Investigation of basic imaging properties of digital radiography. Part 1: modulation transfer function, Med Phys 11, 287-295 (1984).

(19) M. L. Giger, K. Doi and C. E. Metz, Investigation of basic imaging properties of digital radiography. Part 2: noise Wiener spectrum, Med Phys 11, 797-805 (1984).

(20) C. E. Metz, R. F. Wagner, K. Doi, D. Brown, R. M. Nishikawa and K. Myers, Toward consensus on quantitative assessment of medical imaging systems, Med. Phys. 22, 1057-1061 (1995).